/* 
 * Free FFT and convolution (JavaScript)
 * 
 * Copyright (c) 2017 Project Nayuki. (MIT License)
 * https://www.nayuki.io/page/free-small-fft-in-multiple-languages
 * 
 * Permission is hereby granted, free of charge, to any person obtaining a copy of
 * this software and associated documentation files (the "Software"), to deal in
 * the Software without restriction, including without limitation the rights to
 * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
 * the Software, and to permit persons to whom the Software is furnished to do so,
 * subject to the following conditions:
 * - The above copyright notice and this permission notice shall be included in
 *   all copies or substantial portions of the Software.
 * - The Software is provided "as is", without warranty of any kind, express or
 *   implied, including but not limited to the warranties of merchantability,
 *   fitness for a particular purpose and noninfringement. In no event shall the
 *   authors or copyright holders be liable for any claim, damages or other
 *   liability, whether in an action of contract, tort or otherwise, arising from,
 *   out of or in connection with the Software or the use or other dealings in the
 *   Software.
 */

"use strict";


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This is a wrapper function.
 */
function transform(real, imag) {
	var n = real.length;
	if (n != imag.length)
		throw "Mismatched lengths";
	if (n == 0)
		return;
	else if ((n & (n - 1)) == 0)  // Is power of 2
		transformRadix2(real, imag);
	else  // More complicated algorithm for arbitrary sizes
		transformBluestein(real, imag);
}


/* 
 * Computes the inverse discrete Fourier transform (IDFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This is a wrapper function. This transform does not perform scaling, so the inverse is not a true inverse.
 */
function inverseTransform(real, imag) {
	transform(imag, real);
}


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector's length must be a power of 2. Uses the Cooley-Tukey decimation-in-time radix-2 algorithm.
 */
function transformRadix2(real, imag) {
	// Length variables
	var n = real.length;
	if (n != imag.length)
		throw "Mismatched lengths";
	if (n == 1)  // Trivial transform
		return;
	var levels = -1;
	for (var i = 0; i < 32; i++) {
		if (1 << i == n)
			levels = i;  // Equal to log2(n)
	}
	if (levels == -1)
		throw "Length is not a power of 2";
	
	// Trigonometric tables
	var cosTable = new Array(n / 2);
	var sinTable = new Array(n / 2);
	for (var i = 0; i < n / 2; i++) {
		cosTable[i] = Math.cos(2 * Math.PI * i / n);
		sinTable[i] = Math.sin(2 * Math.PI * i / n);
	}
	
	// Bit-reversed addressing permutation
	for (var i = 0; i < n; i++) {
		var j = reverseBits(i, levels);
		if (j > i) {
			var temp = real[i];
			real[i] = real[j];
			real[j] = temp;
			temp = imag[i];
			imag[i] = imag[j];
			imag[j] = temp;
		}
	}
	
	// Cooley-Tukey decimation-in-time radix-2 FFT
	for (var size = 2; size <= n; size *= 2) {
		var halfsize = size / 2;
		var tablestep = n / size;
		for (var i = 0; i < n; i += size) {
			for (var j = i, k = 0; j < i + halfsize; j++, k += tablestep) {
				var l = j + halfsize;
				var tpre =  real[l] * cosTable[k] + imag[l] * sinTable[k];
				var tpim = -real[l] * sinTable[k] + imag[l] * cosTable[k];
				real[l] = real[j] - tpre;
				imag[l] = imag[j] - tpim;
				real[j] += tpre;
				imag[j] += tpim;
			}
		}
	}
	
	// Returns the integer whose value is the reverse of the lowest 'bits' bits of the integer 'x'.
	function reverseBits(x, bits) {
		var y = 0;
		for (var i = 0; i < bits; i++) {
			y = (y << 1) | (x & 1);
			x >>>= 1;
		}
		return y;
	}
}


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
 * Uses Bluestein's chirp z-transform algorithm.
 */
function transformBluestein(real, imag) {
	// Find a power-of-2 convolution length m such that m >= n * 2 + 1
	var n = real.length;
	if (n != imag.length)
		throw "Mismatched lengths";
	var m = 1;
	while (m < n * 2 + 1)
		m *= 2;
	
	// Trignometric tables
	var cosTable = new Array(n);
	var sinTable = new Array(n);
	for (var i = 0; i < n; i++) {
		var j = i * i % (n * 2);  // This is more accurate than j = i * i
		cosTable[i] = Math.cos(Math.PI * j / n);
		sinTable[i] = Math.sin(Math.PI * j / n);
	}
	
	// Temporary vectors and preprocessing
	var areal = newArrayOfZeros(m);
	var aimag = newArrayOfZeros(m);
	for (var i = 0; i < n; i++) {
		areal[i] =  real[i] * cosTable[i] + imag[i] * sinTable[i];
		aimag[i] = -real[i] * sinTable[i] + imag[i] * cosTable[i];
	}
	var breal = newArrayOfZeros(m);
	var bimag = newArrayOfZeros(m);
	breal[0] = cosTable[0];
	bimag[0] = sinTable[0];
	for (var i = 1; i < n; i++) {
		breal[i] = breal[m - i] = cosTable[i];
		bimag[i] = bimag[m - i] = sinTable[i];
	}
	
	// Convolution
	var creal = new Array(m);
	var cimag = new Array(m);
	convolveComplex(areal, aimag, breal, bimag, creal, cimag);
	
	// Postprocessing
	for (var i = 0; i < n; i++) {
		real[i] =  creal[i] * cosTable[i] + cimag[i] * sinTable[i];
		imag[i] = -creal[i] * sinTable[i] + cimag[i] * cosTable[i];
	}
}


/* 
 * Computes the circular convolution of the given real vectors. Each vector's length must be the same.
 */
function convolveReal(x, y, out) {
	var n = x.length;
	if (n != y.length || n != out.length)
		throw "Mismatched lengths";
	convolveComplex(x, newArrayOfZeros(n), y, newArrayOfZeros(n), out, newArrayOfZeros(n));
}


/* 
 * Computes the circular convolution of the given complex vectors. Each vector's length must be the same.
 */
function convolveComplex(xreal, ximag, yreal, yimag, outreal, outimag) {
	var n = xreal.length;
	if (n != ximag.length || n != yreal.length || n != yimag.length
			|| n != outreal.length || n != outimag.length)
		throw "Mismatched lengths";
	
	xreal = xreal.slice();
	ximag = ximag.slice();
	yreal = yreal.slice();
	yimag = yimag.slice();
	transform(xreal, ximag);
	transform(yreal, yimag);
	
	for (var i = 0; i < n; i++) {
		var temp = xreal[i] * yreal[i] - ximag[i] * yimag[i];
		ximag[i] = ximag[i] * yreal[i] + xreal[i] * yimag[i];
		xreal[i] = temp;
	}
	inverseTransform(xreal, ximag);
	
	for (var i = 0; i < n; i++) {  // Scaling (because this FFT implementation omits it)
		outreal[i] = xreal[i] / n;
		outimag[i] = ximag[i] / n;
	}
}


function newArrayOfZeros(n) {
	var result = [];
	for (var i = 0; i < n; i++)
		result.push(0);
	return result;
}